DEMONSTRATIO MATHEMATICA
Vol. XLVII
No 4
2014
Kishor D. Kucche
EXISTENCE AND CONTROLLABILITY RESULTS FOR
MIXED FUNCTIONAL INTEGRODIFFERENTIAL
EQUATIONS WITH INFINITE DELAY
Abstract. Sufficient conditions are established for the existence of solution for mixed
neutral functional integrodifferential equations with infinite delay. The results are obtained
using the theory of fractional powers of operators and the Sadovskii’s fixed point theorem.
As an application we prove a controllability result for the system.
1. Introduction
In this article, we establish existence results of the following mixed functional integrodifferential equations with infinite delay:
d
(1.1)
rxptq ´ gpt, xt qs ` Axptq
dt
ˆ
˙
żt
żb
“ f t, xt , wpt, s, xs qds, hpt, s, xs qds , t P J,
0
(1.2)
0
x0 “ φ P B,
where J “ r0, bs, ´A is the infinitesimal generator of an analytic semigroup
of bounded linear operators T ptq, t ≥ 0 in a Banach space pX, } ¨ }q, the
nonlinear functions g : J ˆB Ñ X, f : J ˆBˆX ˆX Ñ X, w, h : ∆ˆB Ñ X,
∆ “ tpt, sq : 0 ≤ s ≤ t ≤ bu are continuous functions and B is a phase space
defined later. The histories xt : p´8, 0s Ñ X, xt psq “ xpt ` sq, s ≤ 0,
belong to an abstract phase space B.
Due to the application in many areas of applied mathematics, there has
been an increasing interest in the investigation of neutral functional differential and integrodifferential equations with the effect of infinite delay. The
work on first order abstract neutral functional differential equations with
2010 Mathematics Subject Classification: 45N05, 34G20, 34K40, 47D60.
Key words and phrases: mixed equations, fractional powers of operators, infinite delay,
Sadovskii’s fixed point theorem.
DOI: 10.2478/dema-2014-0072
c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology
894
K. D. Kucche
infinite delay was initiated by Hernández and Henríquez in [15, 16]. Afterwards, several works reporting existence results, controllability problem and
other properties of mild solution for first-order abstract neutral functional
differential and functional integrodifferential equations with infinite delay
have been published, see for example, [1]–[6], [10]–[12], [14]–[16], [20] and
the reference cited therein. The problems related to existence, uniqueness
and other properties of mild solutions of mixed functional integrodifferential
equations with bounded delay have been studied in [7, 8, 18, 19] by using
different techniques.
The aim of this paper is to study the existence and controllability of the
problem (1.1)–(1.2) by using the theory of fractional powers of operators and
the Sadovskii’s fixed point theorem. The results of this paper are based on
the analytic semi group theory and the ideas and techniques in Fu [11] and
Li et al. [20]. The results obtained here generalizes the results of [11, 20].
The paper is organized as follows. In Section 2, we recall some preliminaries and list the hypotheses that will be used through out. In Section
3, we establish the existence result for the system (1.1)–(1.2), in Section 4,
we deal with controllability problem and finally in Section 5, an example is
considered to illustrate the application of our result.
2. Preliminaries and hypotheses
We introduce some preliminaries from [9, 13, 17, 22] and hypotheses that
will be used in our further analysis.
We suppose that pX, } ¨ }q is a Banach space, and ´A : Dp´Aq Ñ X
is the infinitesimal generator of a compact analytic semigroup tT ptqut≥0 of
uniformly bounded linear operators. Let 0 P ρpAq. Then it is possible to
define the fractional power Aα , for 0 ă α ≤ 1 , as a closed linear operator
on its domain DpAα q. Furthermore, the subspace DpAα q is dense in X and
}x}α “ }Aα x}, x P DpAα q defines a norm on DpAα q. We denote by Xα
the Banach space DpAα q normed with }x}α .
Lemma 2.1. ([22]) The following properties hold:
(i) If 0 ă β ă α ≤ 1, then Xα Ă Xβ and the imbedding is compact
whenever the resolvent operator of A is compact.
(ii) There exists constant M ≥ 1 such that }T ptq} ≤ M , for all t P J.
(iii) For every 0 ă α ≤ 1 there exists Cα ą 0 such that
Cα
}p´Aqα T ptq} ≤ α , 0 ă t ≤ b.
t
To study the system (1.1)–(1.2), we assume that the histories xt : p´8, 0s
Ñ X, xt pθq “ xpt`θq belong to a seminormed abstract linear space pB, }¨}B q
of functions mapping p´8, 0s into X, which has been introduced by Hale
Existence and controllability results for mixed functional. . .
895
and Kato [13] and widely discussed in [17]. We assume that B satisfies the
following axioms:
(A1) If x : p´8, σ ` bq Ñ X, b ą 0 is continuous on rσ, σ ` bq and xσ P B,
then for every t P rσ, σ ` bq the following statements hold:
(i) xt is in B;
(ii) }xptq} ≤ H}xt }B ;
(iii) }xt }B ≤ Kpt ´ σq supt}xpsq} : σ ≤ s ≤ tu ` M pt ´ σq}xσ }Bα .
Here H ≥ 0 is a constant, K, M : r0, `8q Ñ r0, `8q, Kp¨q is continuous and M p¨q is locally bounded, and H, Kp¨q, M p¨q are independent
of xptq.
(A2) For the function xp¨q in (A1), xt is a B-valued continuous function on
rσ, σ ` bs.
(A3) The space B is complete.
Definition 2.1. A function x : p´8, bs Ñ X is called a mild solution of
the problem (1.1)–(1.2) if x0 “ φ P B on p´8, 0s, the restriction of xp¨q to the
interval J is continuous, and for each s P r0, tq, the function AT pt´sqgps, xs q,
0 ≤ t ă b is integrable and the integral equation
żt
(2.1)
xptq “ T ptqrφp0q ´ gp0, φqs ` gpt, xt q ´ AT pt ´ sqgps, xs qds
0
ˆ
˙
żt
żs
żb
` T pt ´ sqf s, xs , wps, τ, xτ qdτ, hps, τ, xτ qdτ ds, t P J
0
0
0
is satisfied.
Lemma 2.2. ([24], Sadovskii) Let D be a convex, bounded and closed subset
of a Banach space X. If F : D Ñ D is a condensing operator, then F has a
fixed point in D.
We list the following hypotheses in order to establish our existence result.
(H1 ) There exist constants G, Lg ą 0, such that
(i) }Aβ gpt2 , ψ2 q ´ Aβ gpt1 , ψ1 q} ≤ Gp|t2 ´ t1 | ` }ψ2 ´ ψ1 }B q, t1 , t2 P
J, ψ1 , ψ2 P B.
(ii) }Aβ gpt, ψq} ≤ Lg p}ψ}B ` 1q, t P J, ψ P B.
(H2 ) There exist p, q P CpJ, r0, 8qq, such that
›ż t
›
›ż b
›
›
›
›
›
› ωpt, s, ψqds› ≤ pptq}ψ}B and › hpt, s, ψqds› ≤ qptq}ψ}B ,
›
›
›
›
0
0
for every pt, sq P ∆ and ψ P B.
(H3 ) For every positive integer k, there exists αk P L1 pJ, r0, 8qq, such that
sup
}ψ}B ,}x},}y}≤k
}f pt, ψ, x, yq} ≤ αk ptq for a.e. t P J
896
K. D. Kucche
and
ż
1 b
αk psqds “ µ ă 8.
kÑ`8 k 0
(H4 ) (i) For each pt, sq P ∆, the functions ωpt, s, .q, hpt, s, .q : B Ñ X are
continuous and for each ψ P B the functions ωp., ., ψq, hp., ., ψq :
∆ Ñ X are strongly measurable.
(ii) For each t P J, the function f pt, ., ., .q : B ˆ X ˆ X Ñ X is
continuous and for each pψ, x, yq P B ˆ X ˆ X, the function
f p., ψ, x, yq : J Ñ X is strongly measurable.
lim inf
3. Existence result
Theorem 3.1. Let the hypotheses (H1 )–(H4 ) be satisfied. Then the system
(1.1)–(1.2) has a mild solution on p´8, bs if
ˆ
˙
C1´β bβ
´β
L0 :“ GKb }A } `
(3.1)
ă 1,
tβ
„ˆ
˙

C1´β bβ
}A´β } `
(3.2)
Lg ` M ϑµ Kb ă 1,
β
where ϑ “ maxtPJ t1, pptq, qptqu and Kb “ suptKptq : t P r0, bsu.
Proof. Consider the operator Γ defined by
$
’
’
&φptq, t P p´8, 0s,
şt
φqs ` gpt, xt q ´ 0 AT pt ´ sqgps, xs qds
Γpxqptq “ T ptqrφp0q ´ gp0,
¯
´
ş
ş
ş
’
’
%` t T pt ´ sqf s, xs , s wps, τ, xτ qdτ, b hps, τ, xτ qdτ ds, t P J.
0
0
0
Then the equation (2.1) with the initial condition x0 “ φ P B on p´8, 0s
can be written as x “ Γx. We show that Γ has a fixed point xp¨q, then this
fixed point xp¨q is a mild solution of the system (1.1)–(1.2).
Let yp¨q : p´8, bq Ñ X be the function defined by
#
T ptqφp0q, t ≥ 0,
yptq “
φptq,
´8 ă t ă 0.
Then y0 “ φ and the map t Ñ yt is continuous. We can assume that
N “ supt}yt }B : 0 ≤ t ≤ bu.
For each z P CpJ; Xq, zp0q “ 0, we denote by z the function defined by
#
zptq, t P J,
zptq “
0,
´8 ă t ă 0.
If xp¨q satisfies (2.1) with x0 “ φ P C, we can decompose it as xptq “
zptq ` yptq, 0 ≤ t ≤ b, which implies that xt “ z t ` yt for every t P J and the
function zp¨q satisfies
Existence and controllability results for mixed functional. . .
897
żt
zptq “ ´T ptqgp0, φq ` gpt, z t ` yt q ´ AT pt ´ sqgps, z s ` ys qds
0
żt
` T pt ´ sq
0
ˆ
˙
żs
żb
ˆ f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ds, t P J.
0
0
Let F be the operator on Cpr0, bs; Xq defined by
żt
F pzqptq “ ´T ptqgp0, φq ` gpt, z t ` yt q ´ AT pt ´ sqgps, z s ` ys qds
0
żt
` T pt ´ sq
0
ˆ
˙
żs
żb
ˆ f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ds.
0
0
Obviously, the operator Γ having a fixed point is equivalent to F having one,
so it turns out to prove that F has a fixed point.
For each positive integer k, define
Bk “ tz P CpJ; Xq : zp0q “ 0, }zptq} ≤ k, t P Ju.
Then for each k, Bk is clearly convex, bounded and closed subset of CpJ; Xq.
Note that, for any z P Bk and t P J, we have z t ` yt P B. Thus from
assumption (A2)(iii), we have
(3.3)
}z t ` yt }B ≤ }z t }B ` }yt }B
≤ Kptq supt}zpsq} : 0 ≤ s ≤ tu ` M ptq}z 0 }B ` }yt }B
≤ kKb ` N,
since }z 0 }B “ 0. Using Lemma 2.1, hypothesis (H1 (ii)) and the condition
(3.3), we get
}AT pt ´ sqgps, z s ` ys q} ≤ }A1´β T pt ´ sqAβ gps, pzqs ` ys q}
C1´β
≤
Lg pkKb ` N ` 1q.
pt ´ sq1´β
Thus from Bochner Theorem [21], it follows that AT pt ´ sqgps, z s ` ys q is
integrable on r0, tq, and hence, F is well defined on Bk .
We claim that there exists a positive integer k such that F Bk Ď Bk . If
this is not true, then for each positive integer k, there is a function zk P Bk ,
but F pzk q R Bk , that is, }F pzk qptq} ą k for some tpkq P J, where tpkq denotes
898
K. D. Kucche
t depending on k. However, on the other hand, we have
(3.4)
k ă }F pzk qptq}
≤ }T ptq}}A´β }}Aβ gp0, φq} ` }A´β }}Aβ gpt, pz k qt ` yt q}
żt
żt
›
›
› β
›
1´β
T pt ´ sq} ›A gps, pz k qs ` ys q› ds ` }T pt ´ sq}
` }A
0
0
› ˆ
˙›
żs
żb
›
›
›
ˆ ›f s, pz k qs ` ys , wps, τ, pz k qτ ` yτ qdτ, hps, τ, pz k qτ ` yτ qdτ ›› ds.
0
0
Note that, from the condition (3.3) and hypothesis (H2 ), for each t P J, we
have
›ż t
› ›ż b
›*
"
›
› ›
›
(3.5) max }pz k qt ` yt }, ›› wpt, s, pz k qs ` ys qds››, ›› hpt, s, pz k qs ` ys qds››
0
0
≤ pkKb ` N qmaxt1, pptq, qptqu “ pkKb ` N qϑ.
tPJ
Using hypothesis (H1 )–(H3 ) and the inequalities (3.3) and (3.5), from (3.4),
we obtain
k ăM }A´β }Lg p}φ}B ` 1q ` }A´β }Lg pkKb ` N ` 1q
żt
żt
C1´β
αpkKb `N qϑ psqds.
`
L pkKb ` N ` 1qds ` M
1´β g
0 pt ´ sq
0
Therefore,
ˆ
˙
M }A´β }Lg p}φ}B ` 1q
N `1
´β
` }A }Lg Kb `
1ă
k
k
`
˘
N `1
β
C1´β b Lg Kb ` k
`
β
ˆ
˙
żb
N
1
` M Kb `
ϑ
α
psqds.
k
pkKb ` N qϑ 0 pkKb `N qϑ
Taking lower limit and using pH3 q, we get
„ˆ
˙

C1´β bβ
´β
}A } `
Lg ` M ϑµ Kb ≥ 1.
β
This contradicts (3.1). Hence, for some positive integer k, F Bk Ď Bk .
Next, we show that the operator F has a fixed point in Bk by applying
Sadovskii’s fixed point theorem, which implies that (1.1)–(1.2) has a mild
solution. We decompose F as F “ F1 ` F2 , where the operators F1 , F2 are
Existence and controllability results for mixed functional. . .
899
defined on Bk , respectively by
żt
pF1 zqptq “ ´T ptqgp0, φq ` gpt, z t ` yt q ´ AT pt ´ sqgps, z s ` ys qds,
0
żt
pF2 zqptq “
T pt ´ sq
0
ˆ
˙
żs
żb
ˆ f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ds,
0
0
for z P Bk and t P J. To prove F is a condensing operator [24], we will show
that F1 verifies a contraction condition, while F2 is a compact operator.
Firstly, we prove that F1 is a contraction. Take any z1 , z2 P Bk . Then
for each t P J, by Lemma 2.1, hypothesis pH1 qpiq and (3.3), we have
}pF1 z1 qptq ´ pF1 z2 qptq}
≤ }A´β }}Aβ gpt, pz 1 qt ` yt q ´ Aβ gpt, pz 2 qt ` yt q}
żt
›
›
›
›
` }A1´β T pt ´ sq} ›Aβ gps, pz 1 qs ` ys q ´ Aβ gps, pz 2 qs ` ys q› ds
0
żt
C1´β
´β
≤ }A }G}pz 1 qt ´ pz 2 qt }B `
G}pz 1 qs ´ pz 2 qs }B ds
1´β
0 pt ´ sq
≤ }A´β }GtKptq supt}z 1 psq ´ z 2 psq} : 0 ≤ s ≤ tu
` M ptq r}pz 1 q0 }B ` }pz 2 q0 }B su
żt
C1´β
`
GtKpsq supt}z 1 pτ q ´ z 2 pτ q} : 0 ≤ τ ≤ su
1´β
0 pt ´ sq
` M ptq r}pz 1 q0 }B ` }pz 2 q0 }B suds
ˆ
˙
C1´β bβ
´β
≤ GKb }A } `
supt}z1 psq ´ z2 psq} : 0 ≤ t ≤ bu,
tβ
since }pz 1 q0 }B “ 0 and }pz 2 q0 }B “ 0. This implies,
}F1 z1 ´ F1 z2 } ≤ L0 }z1 ´ z2 }.
Therefore, by assumption (3.1), we see that F1 is a contraction.
To prove F2 is a compact operator, first we prove that F2 is continuous
on Bk . Let tzn u Ď Bk with zn Ñ z in Bk ; then for each t P J, pz n qt Ñ z t
and by using hypothesis pH4 q, we have
ˆ
˙
żt
żb
f t, z nt ` yt , wpt, s, pz n qs ` ys qds, hpt, s, pz n qs ` ys qds
0
0
ˆ
˙
żt
żb
Ñ f t, z t ` yt , wpt, s, z s ` ys qds, hpt, s, z s ` ys qds
as n Ñ 8,
0
0
900
K. D. Kucche
for each t P J and since
› ˆ
˙
żt
żb
›
›f t, z nt ` yt , wpt, s, pz n qs ` ys qds, hpt, s, pz n qs ` ys qds
›
0
0
ˆ
˙›
żt
żb
›
´f t, z t ` yt , wpt, s, z s ` ys qds, hpt, s, z s ` ys qds ››
0
0
≤ 2αpkKb `N qϑ ptq,
we have, by dominated convergence theorem
}F zn ´ F z}
›ż t
„ ˆ
żs
›
“ sup›› T pt ´ sq f s, pz n qs ` ys , wps, τ, pz n qτ ` yτ qdτ,
tPJ
0
0
˙
żb
hps, τ, pz n qτ ` yτ qdτ
0
˙ ›
ˆ
żs
żb
›
´f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ds››
0
0
˙
ż t› ˆ
żs
żb
›
›
≤ M ›f s, pz n qs ` ys , wps, τ, pz n qτ ` yτ qdτ, hps, τ, pz n qτ ` yτ qdτ
0
0
0
˙›
ˆ
żs
żb
›
´f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ››ds
0
0
Ñ 0 as n Ñ 8. Therefore, F is continuous.
Next, we prove that the family tF z : z P Bk u is an equicontinuous family
of functions. Let ą 0 be small and 0 ă t1 ă t2 . Then we have
}F2 pzqpt2 q ´ F2 pzqpt1 q}
ż t1 ´
≤
}T pt2 ´ sq ´ T pt1 ´ sq}
0
› ˆ
˙›
żs
żb
›
›
ˆ ››f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ›› ds
0
0
ż t1
`
}T pt2 ´ sq ´ T pt1 ´ sq}
t1 ´
› ˆ
˙›
żs
żb
›
›
›
ˆ ›f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ›› ds
0
0
ż t2
}T pt2 ´ sq}
`
t1
› ˆ
˙›
żs
żb
›
›
›
ˆ ›f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ›› ds
0
0
Existence and controllability results for mixed functional. . .
901
ż t1 ´
≤
}T pt2 ´ sq ´ T pt1 ´ sq}αpkKb `N qϑ psqds
0
ż t1
`
t1 ´
ż t2
`
t1
}T pt2 ´ sq ´ T pt1 ´ sq}αpkKb `N qϑ psqds
}T pt2 ´ sq}αpkKb `N qϑ psqds.
Since T ptq, t ą 0 is compact and hence continuous in the uniform operator
topology, and αpkKb `N qϑ P L1 , we see that }F2 pzqpt2 q ´ F2 pzqpt1 q} Ñ 0
independent of z P Bk as pt2 ´ t1 q Ñ 0 with sufficiently small. This shows
that F maps Bk into an equicontinuous family of functions.
Now, it remains to prove that V ptq “ tF2 pzqptq : z P Bk u is relatively
compact in X. Let 0 ă t ≤ b be fixed and be a real number satisfying
0 ă ă t; for z P Bk , we define
ˆ
ż t´
żs
F2 pzqptq “
T pt ´ sqf s, z s ` ys , wps, τ, z τ ` yτ qdτ,
0
0
˙
żb
hps, τ, z τ ` yτ qdτ ds
0
ˆ
˙
ż t´
żs
żb
“ T pq
T pt´´sqf s, z s `ys , wps, τ, z τ `yτ qdτ, hps, τ, z τ `yτ qdτ ds.
0
0
0
Since
›ż t´
›
›
T pt ´ ´ sq
›
0
˙ ›
ˆ
żs
żb
›
ˆ f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ds››
0
0
ż t´
≤M
αpkKb `N qϑ psqds,
0
the compactness of T ptq pt ą 0q, implies that the set V ptq “ tF2 pzqptq : z P
Bk u is relative compact in X for every , 0 ă ă t. Moreover, by making
use of Lemma 2.1, hypotheses pH2 q, pH3 q and the condition (3.3), for every
z P Bk , we have
}F2 pzqptq ´ F2 pzqptq}
˙›
ˆ
żs
żb
żt ›
›
›
›
›
“
›T pt ´ sqf s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ›ds
t´
0
0
żt
≤
M αpkKb `N qϑ psqds.
t´
902
K. D. Kucche
Therefore, there are relative compact sets arbitrarily close to the set V ptq “
tF2 pzqptq : z P Bk u; hence the set V ptq is also relative compact in X. Thus,
by the Arzela–Ascoli theorem, F2 is a compact operator. These arguments
enable us to conclude that F “ F1 ` F2 is condensing operator on Bk , and
by the fixed point theorem of Sadovskii’s, there exists a fixed point zp¨q for
F on Bk . If we define xptq “ zptq ` yptq, ´8 ă t ≤ b, it is easy to see that
xp¨q is a mild solution of (1.1)–(1.2) satisfying x0 “ φ.
4. Application
As an application of Theorem 3.1, we shall consider the system (1.1)–
(1.2) with control parameter such as:
d
(4.1)
rxptq ´ gpt, xt qs ` Axptq
dt
˙
ˆ
żb
żt
“ Cuptq ` f t, xt , wpt, s, xs qds, hpt, s, xs qds , t P J,
0
0
(4.2)
x0 “ φ P B,
where the control function up¨q is given in L2 pJ, U q - the Banach space of
admissible control functions with U as a Banach space and C is a bounded
linear operator from U into X. The mild solution of the system (4.1)–(4.2)
is given by
żt
(4.3) xptq “ T ptqrφp0q ´ gp0, φqs ` gpt, xt q ´ AT pt ´ sqgps, xs qds
0
żt
` T pt ´ sq
0
„
ˆ
˙
żs
żb
ˆ Cupsq ` f s, xs , wps, τ, xτ qdτ, hps, τ, xτ qdτ ds, t P J,
0
(4.4)
0
x0 “ φ P B.
Definition 4.1. The system (4.1)–(4.2) is said to be controllable on the
interval J if for every initial function φ P B and x1 P X, there exists a
control u P L2 pJ, U q such that the mild solution xp¨q of (4.1)–(4.2) satisfies
xpbq “ x1 .
To establish the controllability result, we need the following additional
hypothesis:
(H6 ) The linear operator W : L2 pJ; U q Ñ X defined by
żb
Wu “
T pb ´ sqCupsqds
0
has an induced inverse operator W̃ ´1 , which takes values in
L2 pJ; U q{kerW .
Existence and controllability results for mixed functional. . .
903
For the construction of the operator W and its inverse, see [23].
Theorem 4.1. Let the hypotheses (H1 )–(H6 ) be satisfied. Then the system
(4.1)–(4.2) is controllable on interval J if
ˆ
˙
C1´β bβ
(4.5)
L0 :“ GKb }A´β } `
ă 1,
tβ
˙

´
¯ „ˆ
C1´β bβ
´1
´β
(4.6)
Lg ` M ϑµ Kb ă 1,
}A } `
1 ` bM }C}}W̃ }
β
where ϑ “ maxtPJ t1, pptq, qptqu and Kb “ suptKptq : t P r0, bsu.
Proof. By use of the hypothesis (H6 ), for an arbitrary function xp¨q, we
define the control
„
żb
´1
x1 ´ T pbqrφp0q ´ gp0, φqs ´ gpb, xb q `
AT pb ´ sqgps, xs qds
uptq “ W̃
0
ˆ
˙ 
żb
żs
żb
´ T pb ´ sq f s, xs , wps, τ, xτ qdτ, hps, τ, xτ qdτ ds ptq.
0
0
0
Using this control, we show that the operator Ψ defined by
$
φptq, t P p´8, 0s,
’
’
ş
’
’
&T ptqrφp0q ´ gp0, φqs ` gpt, xt q ´ t AT pt ´ sqgps, xs qds
0
Ψpxqptq “ ` şt T pt ´ sq
’
’
”0
´
¯ı
’
’
% ˆ Cupsq`f s, x , şs wps, τ, x qdτ, şb hps, τ, x qdτ ds, t P J,
τ
τ
s 0
0
has a fixed point xp¨q. Then this fixed point xp¨q is a mild solution of the
problem (4.1)–(4.2), and we can easily verify that xpbq “ Ψpxqpbq “ x1 . This
means that the control u steers the system from the initial function φ to x1
in time b, which implies that the system is controllable. As in the proof of
Theorem 3.1, the equivalent operator for Ψ is given by
żt
Ωpzqptq “ ´T ptqgp0, φq ` gpt, z t ` yt q ´ AT pt ´ sqgps, z s ` ys qds
0
„
żt
` T pt ´ sq Cupsq
0
ˆ
˙
żs
żb
` f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ds.
0
0
Our aim is to prove that there exists a positive integer k such that ΩBk Ď Bk .
If possible, for each positive integer k, there is a function zk P Bk such that
Ωpzk q R Bk , then }Ωpzk qptq} ą k for some t depending on k. But by using
the hypotheses (H1 )–(H3 ), Lemma 2.1 and the inequalities (3.3) and (3.5),
904
K. D. Kucche
we obtain
k ă }Ωpzk qptq}
(4.7)
≤ M }A´β }Lg p}φ}B ` 1q ` }A´β }Lg pkKb ` N ` 1q
żt
żt
C1´β
L pkKb ` N ` 1qds ` M }C}}uk psq}ds
`
1´β g
0
0 pt ´ sq
żt
`M
αpkKb `N qϑ psqds,
0
where uk is corresponding control of xk , xk “ z k ` y. Note that
(4.8)
}uk psq}
"
´1
≤ }W̃ } }x1 } ` }T pbq}r}φ}B ` }A´β }}Aβ gp0, φq}s
` }A´β }}Aβ gpb, pz k qb ` yb q}
żb
›
›
›
›
`
}A1´β T pb ´ τ q} ›Aβ gpτ, pz k qτ ` yτ q› dτ
0
› ˆ
żb
›
`
}T pb ´ τ q}››f τ, pz k qτ ` yτ ,
0
żτ
żb
˙› *
›
hpτ, σ, pz k qσ ` yσ qdσ ››dτ
wpτ, σ, pz k qσ ` yσ qdσ,
0
0
"
≤ }W̃ ´1 } }x1 } ` M r}φ}B ` }A´β }Lg p}φ}B ` 1qs ` }A´β }Lg pkKb ` N ` 1q
*
żb
żb
C1´β
L pkKb ` N ` 1qdτ ` M
αpkKb `N qϑ pτ qdτ .
`
1´β g
0
0 pb ´ τ q
Using the estimate (4.8) in (4.7), we get
(4.9)
k ă M }A´β }Lg p}φ}B ` 1q ` }A´β }Lg pkKb ` N ` 1q
C1´β bβ Lg pkKb ` N ` 1q
β
"
´1
` bM }C}}W̃ } }x1 } ` M r}φ}B ` }A´β }Lg p}φ}B ` 1qs
`
C1´β bβ Lg pKb ` N ` 1q
` }A´β }Lg pkKb ` N ` 1q
β
*
żb
żb
`M
αpkKb `N qϑ psqds ` M
αpkKb `N qϑ psqds
0
0
Existence and controllability results for mixed functional. . .
905
“ M }A´β }Lg p}φ}B ` 1q
!
)
` bM }C}}W̃ ´1 } }x1 } ` M r}φ}B ` }A´β }Lg p}φ}B ` 1qs
` p1 ` bM }C}}W̃ ´1 }q}A´β }Lg pkKb ` N ` 1q
C1´β bβ Lg pkKb ` N ` 1q
β
żb
` p1 ` bM }C}}W̃ ´1 }qM
αpkKb `N qϑ psqds.
` p1 ` bM }C}}W̃ ´1 }q
0
Therefore,
ˆ
˙ ˆ
˙
C1´β bβ
M˚
N `1
´1
´β
1ă
` p1 ` bM }C}}W̃ }q }A } `
Lg Kb `
k
β
k
ˆ
˙
żb
N
1
` p1 ` bM }C}}W̃ ´1 }qM Kb `
α
psqds,
ϑ
k
pkKb ` N qϑ 0 pkKb `N qϑ
where
M ˚ “ M }A´β }Lg p}φ}B ` 1q
!
)
` bM }C}}W̃ ´1 } }x1 } ` M r}φ}B ` }A´β }Lg p}φ}B ` 1qs
is independent of k. Taking lower limit and using pH3 q, we get
˙

´
¯ „ˆ
C1´β bβ
´1
´β
1 ` bM }C}}W̃ }
}A } `
Lg ` M ϑµ Kb ≥ 1.
β
This contradicts (4.6). Hence, for some positive integer k, we must have
ΩBk Ď Bk .
In order to apply Sadoviskii’s fixed point theorem, we decompose Ω as
Ω “ F1 ` Ω1 , where the operators F1 , Ω1 are defined on Bk , respectively by
żt
F1 pzqptq “ ´T ptqgp0, φq ` gpt, z t ` yt q ´ AT pt ´ sqgps, z s ` ys qds,
0
„
żt
Ω1 pzqptq “
T pt ´ sq Cupsq
0
ˆ
˙
żs
żb
` f s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ds
0
0
906
K. D. Kucche
żt
„
T pt ´ sqC W̃
“
´1
x1 ´ T pbqrφp0q ´ gp0, φqs ´ gpb, z b ` yb q
0
żb
`
0
żb
´
AT pb ´ τ qgpτ, z τ ` yτ qdτ
ˆ
żτ
wpτ, σ, z σ ` yσ qdσ,
T pb ´ τ qf τ, z τ ` yτ ,
0
0
żb
˙ 
hpτ, σ, z σ ` yσ qdσ dτ psqds
0
ˆ
˙
żs
żb
żt
` T pt ´ sqf s, z s ` ys , wps, τ, z τ ` yτ qdτ, hps, τ, z τ ` yτ qdτ ds,
0
0
0
for z P Bk and t P J. We have already proved that F1 verifies a contraction
condition. The proof that Ω1 is a compact operator can be completed by
closely looking at the proof of compactness of the operator F2 .
5. Example
Consider the mixed partial differential equation system
żt żπ
ı
B”
(5.1)
zpt, wq `
bps ´ t, y, wqzps, yq dy ds
Bt
´8 0
żtżs
B2
“
zpt, wq ` apwqzpt, wq `
cpσ ´ s, y, wqzpσ, wq dσ ds
Bw2
0 ´8
żbżs
`
dpσ ´ s, y, wqzpσ, wq dσ ds, 0 ≤ t ≤ b, 0 ≤ w ≤ π,
0
(5.2)
(5.3)
´8
zpt, 0q “ zpt, πq “ 0,
zpθ, wq “ φpθ, wq, θ ≤ 0,
0 ≤ w ≤ π.
To write the system (5.1)–(5.3) in the form (1.1)–(1.2), we shall take X “
L2 r0, πs, and define xptq “ zpt, ¨q The operator A is defined by Av “ ´v 2
with the domain
DpAq “ tv P X : v, v 1 absolutely continuous, v 2 P X, vp0q “ vpπq “ 0u.
It is well known that ´A generates a strongly continuous semigroup tT ptqut≥0
which is analytic, compact and self-adjoint. Furthermore, ´A has a discrete
spectrum, the eigenvalues b
are ´n2 , n P N , with the corresponding normalized eigenvectors zn pxq “
(a) If v P DpAq then Av “
2
π
sinpnxq. Then the following properties hold:
ř8
n“1 n
2 xv, z yz .
n n
Existence and controllability results for mixed functional. . .
907
(b) For every v P X,
8
ÿ
T ptqv “
A´1{2 v “
2
e´n t xv, zn yzn ,
n“1
8
ÿ
1
xv, zn yzn .
n
n“1
In particular, }T ptq} ≤ e´t and }A´1{2 } “ř1.
(c) The operator A1{2 is given by A1{2 v “ 8
n“1 nxv, zn yzn , on the space
ř
nxv,
z
yz
P
Xu.
DpA1{2 q “ tv P X, 8
n n
n“1
Consider the phase space B “ Cr ˆ Lp pg; Xq, r ≥ 0, 1 ≤ p ≤ 8, with r “ 0,
p “ 2 and X “ L2 r0, πs, which has been discussed in [17]. Assume that the
following conditions hold:
(i) The function bp¨q is measurable, and
żπż0 żπ´
¯
b2 pθ, y, wq{gpθq dy dθ dw ă 8.
0
´
(ii) The function
and
´8 0
¯
B
Bw bpθ, y, wq is measurable, bpθ, y, 0q “ bpθ, y, πq “ 0,
żπż0 żπ
¯2
1 ´ B
bpθ, y, wq dy dθ dw ă 8.
0 ´8 0 gpθq Bw
ş0
(iii) The function cp¨q is measurable with ´8 pc2 pθq{gpθqqdθ ă 8.
ş0
(iv) The function dp¨q is measurable with ´8 pd2 pθq{gpθqqdθ ă 8.
(v) The function φ defined by φpθqpwq “ φpθ, wq belongs to B.
Define the functions g : J ˆ B Ñ X, f : J ˆ B ˆ X ˆ X Ñ X, w, h :
∆ ˆ B Ñ X by
ż0 żπ
gpt, ψqpξq “
bpθ, y, wqψpθ, wq dw dθ,
´8 0
ż0
wpt, s, ψqpξq “
cpθ, y, wqψpθ, wqdθ,
´8
ż0
hpt, s, ψqpξq “
dpθ, y, wqψpθ, wqdθ,
´8
żt
ˆ
f
t, ψ,
żb
kpt, s, ψqds,
0
˙
hpt, s, ψqds pξq
0
żt
“ apξqψp0, ξq `
żb
kpt, s, ψqpξqds `
0
hpt, s, ψqpξqds.
0
908
K. D. Kucche
Under the conditions (i) to (v), the above defined functions verifie the hypothesis of the Theorem 3.1 (see [11, 20] for details). Additionally, if (3.1)
and (3.2) hold, the system (5.1)–(5.3) has mild solution on r´r, 8q.
Acknowledgments. The author is grateful to University Grants Commission, New Delhi, India, for the funding through Minor Research Project
(F. No. 42-997/2013(SR)). The author would like to thank anonymous referee for his/her helpful comments.
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K. D. Kucche
DEPARTMENT OF MATHEMATICS
SHIVAJI UNIVERSITY
KOLHAPUR-416 004, MAHARASHTRA, INDIA
E-mail: [email protected]
Received June 3, 2013; revised version August 12, 2013.
Communicated by A. Fryszkowski.